### Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test

**Authors:** Elemér E Rosinger

**Comments:** 184 pages

It is shown how the infinity of differential algebras of generalized
functions is naturally subjected to a basic dichotomic singularity test
regarding their significantly different abilities to deal with large classes
of singularities. In this respect, a review is presented of the way
singularities are dealt with in four of the infinitely many types of
differential algebras of generalized functions. These four algebras, in the
order they were introduced in the literature are : the nowhere dense,
Colombeau, space-time foam, and local ones. And so far, the first
three of them turned out to be the ones most frequently used in a
variety of applications. The issue of singularities is naturally not a
simple one. Consequently, there are different points of view, as well as
occasional misunderstandings. In order to set aside, and preferably,
avoid such misunderstandings, two fundamentally important issues
related to singularities are pursued. Namely, 1) how large are the sets
of singularity points of various generalized functions, and 2) how are
such generalized functions allowed to behave in the neighbourhood of
their point of singularity. Following such a two fold clarification on
singularities, it is further pointed out that, once one represents
generalized functions - thus as well a large class of usual singular functions
- as elements of suitable differential algebras of generalized functions,
one of the main advantages is the resulting freedom to perform
globally arbitrary algebraic and differential operations on such functions,
simply as if they did not have any singularities at all. With the same
freedom from singularities, one can perform globally operations such
as limits, series, and so on, which involve infinitely many generalized
functions. The property of a space of generalized functions of being
a flabby sheaf proves to be essential in being able to deal with large
classes of singularities. The first and third type of the mentioned
differential algebras of generalized functions are flabby sheaves, while the
second type fails to be so. The fourth type has not yet been studied
in this regard.

**Category:** Functions and Analysis